In this article, we present an approximation algorithm for solving the single source shortest paths problem on weighted polyhedral surfaces. We consider a polyhedral surface P as consisting of n triangular faces, where each face has an associated positive weight. The cost of travel through a face is the Euclidean distance traveled, multiplied by the face's weight. For a given parameter ε, 0 < ε < 1, the cost of the computed paths is at most 1 + ε times the cost of corresponding shortest paths. Our algorithm is based on a novel way of discretizing polyhedral surfaces and utilizes a generic greedy approach for computing shortest paths in geometric graphs obtained by such discretization. Its running time is O(C(P) n √ ε log n ε log 1 ε ) time, where C(P) captures geometric parameters and the weights of the faces of P.
We consider the classical geometric problem of determining a shortest path through a weighted domain. We present approximation algorithms that compute e-short paths, i.e., paths whose costs are within a factor of 1 + e of the shortest path costs, for an arbitrary constant e > O, for the following geometric configurations: SPPS Problem:
This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any n-vertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( p (g + 1=")n) vertices. Our result improves the best previous ones with respect to the size of C and the time complexity of the algorithm. Moreover, we show that one can cut o from G a piece of no more than (1 ?")n vertices by removing a set of O( p n"(g" + 1) vertices. Both results are optimal up to a constant factor.
We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P , where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the sub-paths within each face of P .We present query algorithms that compute approximate distances and/or approximate shortest paths on P . Our all-pairs query algorithms take as input an approximation parameter ε ∈ (0, 1) and a query time parameter q, in a certain range, and builds a data structure APQ(P, ε; q), which is then used for answering ε-approximate distance queries in O(q) time. As a building block of the APQ(P, ε; q) data structure, we develop a single source query data structure SSQ(a; P, ε) that can answer ε-approximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.
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